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Lattice QCD and the standard model
UCLEAR PHYSIC~
EI.SEVIER
Nuclear Physics B (Proc. Suppl.) 49 (1996) 269581
PROCEEDINGS SUPPLEMENTS
Lattice QCD and the standard model Andreas S. Kronfeld ~ ~Theoretieal Physics Group, Fermi National Accelerator Laboratory, Batavia, Illinois, USA Most of the poorly known parameters of the Standard Model cannot be determined without reliable calculations in nonperturbative QCD. Lattice gauge theory provides a first-principles definition of the required functional integrals, and hence offers ways of performing these calculations. This paper reviews the progress in computing hadron spectra and electroweak matrix elements needed to determine ~xs, the quark masses, and the CabibboKobayashi-Maskawa matrix.
I. I N T R O D U C T I O N Many reviews of elementary particle physics start by celebrating (or lamenting!) the success of the Standard Model. Indeed, with some eighteen or nineteen parameters the SU(3)xSU(1)xU(1) gauge theory explains an enormous array of experiments. Even a terse compendium [1] of the experiments is more than big enough to fill a phone book. A glance at Table 1 shows, however, that roughly half of the parameters are not so well determined. To test the Standard Model stringently one must learn the values of these parameters more precisely. Most of this volume contains abstract theoretical physics. Often the motivation of such work is to deduce a more appealing and more unified theory of particle physics. It might be possible to find better theories by relying on aesthetics and mathematical consistency; Einstein's general relativity is often cited as an example. Nevertheless, any natural science must have recourse to some experimental data. For a model based on superstrings, or whatever else, one could take the view that Table 1 represents the data. Then it is alarming that many of the most central data are known only within a factor of three or four. Under these circumstances, it will be difficult to know if beauty found in mathematics is the same as beauty found in nature. Except for the mass of the Higgs boson (or any other undiscovered remnant of electroweak symmetry breaking), the poorly known parameters all involve quarks. Other than top [2], which de-
cays too quickly for confinement to play a role, the masses of the quarks are a bit better than wild guesses. The information on the CabibboKobayashi-Maskawa (CKM) quark-mixing matrix is spotty, especially when one relaxes the assumption of three-generation unitarity, as shown in Table 2. They are poorly determined simply because experiments measure properties not of quarks, but of the hadrons inside which they are confined. Of course, everyone knows what to do: calculate with QCD, the part of the Standard Model that describes the strong interactions. But the strong coupfing a s is known only at the 5% level; not bad, but nothing fike the fine structure or Fermi constants. Moreover, the binding of quarks into hadrons is nonperturbative-even knowing o~s, hadronic calculations cannot be done on the back of an envelope. The most systematic technique for understanding nonperturbative QCD is lattice gauge theory. The lattice provides quantum field theory with a consistent and mathematically well-defined ultraviolet regulator. At fixed lattice spacing, the quantities of interest are straightforward (combinations of) functional integrals. These integrals can be approximated by a variety of techniques borrowed from statistical mechanics. Especially promising is a numerical technique, the Monte Carlo method with importance sampling, which has become so pre-eminent that the young and uninitiated probably haven't heard of any other. Results from lattice-QCD Monte Carlo calculations have begun to influence Table 1. The world average for the SU(3) gauge coupling a s includes
0920-5632/96/$15.00 -~ 1996 Elsevicr Scicnce B.V. All rights reserved. PII: S0920-5632(96)00344-1
270
A.S. KronfeM/Nuclear Physics B (Proc. Suppl.) 49 (1996) 26~281
Table 1 Parameters of the standard model and lattice calculations that will help determine them. Numerical values taken from the 1994 Review of Particle Properties [1], except m t (ref. [2]), sin 6, and 0QCD. The strong coupling a s refers to the MS scheme at M z . Adapted from ref. [3]. what
value or range
how
gauge couplings O%m
1/137.036 1.166 GeV -2 0.116 + 0.005
105GF as electroweak m a s s e s mz 91.19 GeV
AmlP-lS
what
value or range
IV ,, l
0.974
IVy,, [
0.218-0.224 0.002-0.005 0.180-0.228 0.800-0.975 0.032-0.048 0.0-0.13
IVy,b[ [Vca[
]Vc,[ [V~s[
> 58 GeV
IV,al Iv.I
0.0-0.56
0.51100 MeV 105.66 MeV 1777 MeV
IV, hi
0.0-0.9995
mH lepton m a s s e s
me mu m~.
Table 2 Ranges for CKM matrix elements [Vq~] assuming unitarity but not three generations. Numerical values taken from the 1994 Review of Particle Properties. In three generations IVy,d[ = s12, Iv~bl -- s=3, and [Vub[ = 813, to excellent approximation. how K --* Trey B --, flu D --, f l y D ---, K l u B -~ D * l v f~BB; BK
f .Bs.
quark m a s s e s
rn~ m d
m, mc
2-8 MeV 5-15 MeV 100-300 MeV 1.0-1.6 GeV 4.1-4.5 GeV 180 ± 14 GeV
2 m~, m ~ m~
0.218-0.224 0.032-0.048 0.002-0.005
K ---* ~rev B --* D* lu B ---* ~rlv BK, BB, BB.
mb mt CKM matrix
st2 s2a sta sin 6
#0
msl¢
mi
Q C D v a c u u m angle
0QCD
< 10 -9
dn
results from lattice calculations of the quarkonium spectrum [4-7]. The same calculations are also providing some of the best information on the charm [8] and b o t t o m [9] masses. This is an auspicious beginning. Over the next several years the lattice Q C D calculations will mature. They will help to determine the other unknowns--light quark masses and the C K M matrix. The third column of Tables I and 2 listsrelevant quantities or processes, and the rest of this talk explains how
the program fits together. Sect. 2 gives the non-expert some Perspective on the conceptual and numerical strengths and weaknesses of lattice QCD. Sect. 3 reviews 1) the status of the light hadron spectrum and the prospects for extracting mu, rod, and m , ; and 2) results for the quarkonium spectrum, which yield a s , rob, and me. Sect. 4 outlines lattice QCD calculations of eleetroweak, hadronic matrix elements that are needed to pin down the unitarity triangle of the CKM matrix. There are, of course, m a n y other interesting applications to electroweak phenomenology; for more comprehensive reviews the reader can consult some of the papers listed in the bibliography [3,10]. 2. R U D I M E N T S
OF LATTICE
GAUGE
THEORY In quantum field theory physical measurements are related to vacuum expectation values ((9). Feynman's functional integral representation is
((9)= L--oolim=--0nmZ-lfHaA"(~)× *," H d¢~ (z)d¢~(z) (9 e - s ( a " ' ¢ ' • ) ,
(1)
271
A.S. KronfeM/Nuclear Physics B (Proc. Suppl.) 49 (1996) 26~281
where S is the action--in our case the action of QCD, so it depends on the gluons A~ and the quarks ¢ and anti-quarks ¢. The observable (9 depends on the physics under investigation. The integration over all components and positions of the basic fields, with weight e i s would reproduce the familiar SchrSdinger or Heisenberg formulations of quantum mechanics. The more convergent weight e - s provides some benefits and imposes some restrictions--see below. The limits are there for mathematical rigor; without them the integrals are not well defined. These "cutoffs" also have a physical significance: we do not claim to understand physics either at distances smaller than a (the ultraviolet cutoff), or at distances larger than L (the infrared cutoff). The limit a ~ 0 requires the renormalization group; it must be carried out holding L and physical, infrared scales fixed. In particular, the integration variables At,(z), ¢ ( z ) , and ¢ ( z ) really represent all degrees of freedom in a block of size a 4. The limit "a ---* 0" can be obtained not only hterally, but also by improving the action of the blocked fields. These observations apply to any cutoff scheme for quantum field theory. A nice introduction to the renormalization-group aspects is a summerschool lecture by Lepage [11]. In lattice gauge theory a is nothing but the spacing between lattice sites. If there are N on a side, L = N a . For given N one can compute the integrals numerically. With the 10z-101°-dimensional integrals that arise, the only viable technique is a statistical one: Monte Carlo with importance samphng. In the last few years Monte Carlo calculations have improved in m a n y ways. An especially keen development is the use of improved actions. By universality any action
(2) n
with arbitrary interactions S,~ built with the right symmetry out of the appropriate fields, should have the same continuum bruit. But the cutoff artifacts can be reduced systematically by choosing the "irrelevant" couplings as functions of the "relevant" one. In the case of QCD the relevant coupfings are the gauge couphng and (dimension-
less) quark masses. A so-called renormalized trajectory c°
(Z)
--
has no cutoff effects. Making an error 6Cn in the couphng of the Monte Carlo's action leads to artifacts .-~ 6c,~(S,~), so the objective is to estimate the eouphngs cn of the most important interactions S,~. Of course, this strategy is old, going back to Wilson [12,13] and to Symanzik [14]. New are more effective ways [15-17] of computing the couphngs cn(g2,rnqa), so that cutoff artifacts are genuinely reduced. In particular, for massive quarks, such as charm and bottom, it is wise [18] to treat exactly the mass dependence of the renormahzed trajectory prescribed by eq. (3). With good estimates of the c,~, the cutoff effects can be reduced below other relevant uncertainties. To compute masses one takes an observable O = ff(t)ff*(0), where if(t) is an operator at time t with the flavor and angular-momentum quantum numbers of the state of interest. One can eonstruet such operators using s y m m e t r y alone. (The radial quantum number would require a solution of the theory, but that's what we're after.) The two-point function (~(t)~*(0))
=~
(4)
e - m " ' I(01~l n) 12 ,
n
where the sum is over the radial q u a n t u m number. The exponentials are a happy consequence of the weight e - s in eq. (1). It is advantageous because at long times t only the lowestlying state survives. In a numerical calculation masses are obtained by fitting two-point functions, once single-exponential behavior is verified. Since • is largely arbitrary, some artistry enters: if single-exponential behavior sets in sooner, the statistical quality of the mass estimate is better. Most of Table 1 requires a matrix element of part of the electroweak Hamiltonian 7-/. For the transition from hadron "B" to hadron "Tr" one employs the observable O = ¢b~(t,~)7~(th)fft(o). At long times th and t , - t h the three-point function (ff.(t,)~(th)~(0))
~
e -m'(t'-t~)-rnBta
(01¢. I~r)0rlT~lB)(Bl~ 10),
x
(5)
A.S. Kronfeld/Nuclear Physics B (Proc. Suppl.) 49 (1996) 26~281
272
plus excited-state contributions. If, as in decays of hadrons to leptons, the hadronic final-state is the vacuum, a two-point function will do:
(~(t)O~(O)) = ~ The
e-~"t(O['}/[B,~)(B,~IO~]O>.
(6)
desired
matrix elements (~r[7~[B) and be obtained from eq. (5) and (6), because the masses and matrix elements of eh are obtained from eq. (4). To obtain good results from eqs. (4)-(6), it is crucial to devise nearly optimal operators in the two-point analysis. Consumers of numerical results from lattice QCD should be wary of results, still too prevalent in the literature, that are contaminated by unwanted states. Fortunately the goal of improving the hadronic part of Table 1 can be met with two- and threepoint functions only. Four-point functions are much more demanding technically. In the numerical work that mostly concerns us here, the integrals are computed at a sequence of fixed a's and L's. One adopts a standard mass, say rap, and defines
(oI~IB> can
masses. From the renormalization group, the adjustment of the gauge coupling is equivalent to setting the lattice spacing in physical units, eq. (7). Once the parameters are determined by 1 + n! experimental inputs, QCD should predict all other strong-interaction phenomena. There is no need to introduce condensates, as in I T E P sum rules, or non-renormalizable couplings, as in chiral perturbation theory or heavy-quark effective theory. If theory and experiment disagree, it is a signal of new physics. There are some disadvantages. A practical, though not conceptual, problem is that largescale computational work is more labor-intensive than traditional theoretical physics. Careful work is needed to estimate the uncertainties reliably. Recent improvements in computer power and algorithms have helped practitioners understand their uncertainties better and better. As the consumers of their results become commensurately sophisticated, this trend will continue. After all, in the context of Table 1, meaningful error bars are just as important as the central value.
(amp) 1QOD a -
oxp,
(7)
mp
to obtain the lattice spacing in physical units, and other quantities are predicted via (amB)lq cD mB
-
(s)
a
For continuum-limit, infinite-volume results this is the same as extrapolating dimensionless ratios,
mB mp
lim lira L--oo a - - O
amp(a, L)
.
(9)
There is theoretical guidance for both limits. According to general properties of massive quant u m field theories in finite boxes [19], the infinitevolume limit is rapid for m,~L >> 1, exponential for masses. In non-Abelian gauge theories the renormalization-group a ---* 0 limit is controlled by asymptotic freedom. The main strength of lattice QCD is that it is QCD. It has 1 + n! adjustable parameters, corresponding to the gauge coupling and the quark
2.1. T h e q u e n c h e d a p p r o x i m a t i o n The biggest disadvantage of most of the numerical results mentioned in this talk is something called the "quenched" approximation. A meson consists of a valence quark and anti-quark exchanging any number ofgluons. The gluons can turn into virtual quark loops and back again. The latter process costs a factor of 100-1000 in computer time, so many Monte Carlo programs just omit the virtual quark loops. Borrowing a jargon from condensed-matter physics, this is most often called the quenched approximation. If quenched QCD makes any sense, it is as a kind of model or effective theory. The parameters of quenched QCD can be tuned to reproduce physics at one scale. But the/3 function of quenched and genuine QCD differ, as one sees in perturbation theory, so one cannot expect agreement at all scales. As with any model, only in special cases can one argue that these effects are correctable or negligible; these cases will be highlighted in the rest of the talk.
A.S. KronfeM/Nuclear Physics B (Proc. Suppl.) 49 (1996) 26~281 2.5 GFI 1 experiment
t
t.
ft
t
1.5
t" i
I
K* 0
I
L
I
I
i
N-+E-NA Z* E* f~ 10fitlOfKfK/~t~t
Figure 1. The spectrum and decay constants of the hght hadrons. Error bars are from lattice calculations in the quenched approximation [20,21], and • denotes experiment.
3. Q C D C O U P L I N G S : MASSES
as AND
QUARK
3.1. L i g h t h a d r o n s a n d l i g h t - q u a r k m a s s e s Over the past few years a group at IBM has carried out a systematic calculation of the hghthadron spectrum using the dedicated supercomputer G F l l [20]. They have numerical d a t a for 5 different combinations of (a, L). At L m, 2.3 fm there are three lattice spacings varying by a factor of ~ 2.5. At the coarsest lattice spacing (a -1 ~ 1.4 GeV) there are three volumes, up to almost 2.5 fm. A variety of quark masses are used, and the physical strange quark is reached by interpolation, but the hght (up and down) quarks through extrapolation. The mass dependence is assumed linear, as expected from weakly broken chiral symmetry, and the d a t a substantiate the assumption. Units in MeV have been fixed with mp and the quark masses with m~ and m g . The final results, after extrapolation to the continuum limit and infinite volume, are shown in fig. 1 for two vector mesons and six baryons. (The quark-mass interpolation could reach only the combination mE + m s - mN.) Despite the quenched approximation the agreement with experiment is spectacular.
273
Fig. 1 also includes results from the same investigation for decay constants. The agreement of f,~/mp and fIc/mp is not as good as for the masses. Because of the quenched approximation, this is not entirely unexpected. Recall the argument concerning distance scales and effective theories in sect. 2.1. The binding mechanism responsible for the masses encompasses distances out to the typical hadronic radius. The decay constant, on the other hand, is proportional to the wavefunetion at the origin and thus is more sensitive to shorter distance scales. One sees better agreement when forming the ratio fg/f,~, which--recall eq. (7) and subsequent discussion-is like retuning to the shorter distance. One would like to use the hadron masses to extract the quark masses. Because of confinement, the quark mass more like a renormallzed couphng than the classical concept of mass. Calculations like the one described above yield immediately the bare mass of the lattice theory. More useful to others would be the MS scheme of dimensional regularization. A one-loop perturbative calculation can be used to convert from one scheme to another [22-24,8]. For the light quarks it is convenient to discuss the combinations ~ = ~(md + rn~), Am~,, = m ] - m~, 2 and ms Ratios of the light-quark masses are currently best estimated using chiral perturbation theory [25]. To set the overall scale requires a dynamical calculation in QCD. In lattice QCD, ~h and m, can be extracted from the variation in the square of the pseudoscalar mass between m r2 and m~c. The most difficult quark-mass combination is A m ~ , which causes the isospin-violating contribution to the sphttings in hadron multiplets. Since chiral perturbation theory provides a formula for Am~,/m2, with only second-order corrections, it is likely that the best determination of A m ~ will come from combining the formula with a lattice QCD result for m~. Using the compilation of quenched and unquenched results of Ukawa [26], Mackenzie [27] has estimated thug(1 GeV) ~ 2.3 MeV and m ,~-g(1 GeV) ,~ 65 MeV. The symbol -~ stresses the lack of error bar. This is outside the ranges of 3.5-11.5 MeV and 100-300 MeV indicated in Table 1. A more recent analysis of the strange
274
A.X Kronfeld/Nuclear Physics B (Proc. Suppl.) 49 (1996) 269 281
quark finds m , ~ ( 2 GeV) = 127 ± 18 MeV [28], in the lower part of the range in Table 1. None of these results should be taken seriously until a more complete error analysis exists, but it is intriguing that the conventional estimates might be too high. Q u a r k o n i a : a s , me, a n d m b Quarkonia are bound states of a heavy quark and heavy anti-quark. Three families of states exist, charmonium (7/¢, J / ¢ , etc), bottomonium (~/b, T, etc), and the as yet unobserved Be (b~ and bc bound states). Compared to light hadrons, these systems are simple. The quarks are nonrelativistie, and potential models give an excellent empirical description. But a fundamental treatment of these systems requires nonperturbative QCD, i.e. lattice QCD. Potential models can be exploited, however, to estimate lattice artifacts, and in the quenched approximation they can be used to make corrections. Many states have been observed in the lab, providing cross-checks of the methodology of uncertainty estimation. Once the checks are satisfactory, one can use the spectra to determine a s , m~, and ms. One can also have some confidence in further applications, such as the phenomenology of D and B mesons discussed in sect. 4. For charm, and especially for bottom, the quark mass is close to the ultraviolet cutoff, 1/a or ~r/a, of present-day numerical calculations. Originally lattice gauge theory was formulated assuming rnqa << 1, so quarks mqa ~-- 1 require some reassessment. There are four ways to react. The patient, stolid way is to wait ten years, until computers are powerful enough to reach a cutoff of 20 G e V - - n o t very inspiring. The naive way is to extrapolate from smaller masses, assuming the mqa << 1 interpretation of the lattice theory is adequate; history shows that naive extrapolations can lead to naive and, thus, unacceptable error estimates. The insightful way is to formulate an effective theory for heavy quarks with a lattice cutoff [29,30]; this is the computationally most efficient approach, and when the effectiveness of the heavy-quark expansion is a priori clear, it is the method of choice. The compulsive way to examine a wide class of lattice theories without 3.2.
> CD 0.5
-
0.0
-
D3
I
77c
J/~
hc
Xc0
Xcl
Xcz
Figure 2. A comparison of the charmonium spectrum as calculated in lattice QCD, using two different methods, ref. [31]: o, ref. [32]: D. From
ref. [6].
assuming either mqa << 1 or mq >> (AQcD, a - l ) ; by imposing physical normalization conditions on masses and matrix elements, it is possible to interpret the correlation functions at any value of mqa [18]. The underlying reason is that the lattice theory is completely compatible with the heavy-quark hmit, so the mass-dependent interpretation connects smoothly onto both the insightful method for mqa >> 1 and the standard method for mqa << 1. Fig. 2 shows the charmonium spectrum, on a scale appropriate to the spin-averaged spectrum. Light quark loops are quenched in these calculations [31,32]. The agreement with experimental measurements is impressive, but fig. 2 barely displays the attainable precision. Fig. 3 shows the fine and hyperfine structure of the P states, now for bottomonium [31]. (The 1P1 state hb has not been observed in the lab; the hc has been seen.) The authors of ref. [31] also have results with the virtual quark loops from two light quarks, i.e. up and down are no longer quenched, but strange still is. The agreement is comparable [33]. To obtain these results only two parameters have been adjusted. The standard mass in eq. (7) is AmlP-lS, the spin-averaged splitting of the 1P and 1S states, which is insensitive to the quark mass. By the renormahzation group, this is equivalent to ehminating the bare gauge cou-
A.S. Kronfeld/Nuclear Physics B (Proc. Suppl.) 49 (1996) 269 281 MeV
using
20--+--Xb2 a-0
* hb
--~--Xbl
--20 --40
275
--t-- Xbo
Figure 3. Lattice QCD results for the spindependent splittings of the lowest-lying P states in bottomonium. The dashed lines are the experimental values, where available. Energies are measured relative to the spin average of the states Xbj" From ref. [31].
aAmlP-lS 460 MeV '
where the numerator is the 1P-1S splitting in lattice units. Steps 1 and 4 are explained above. Step 2 requires one-loop perturbation theory, suitably optimized [15]. Step 3 is crucial, because without it the results have no business in Table 1. Consider first a s , and recall the idea of treating the quenched approximation as an effective theory. One sees that the couplings are implicitly matched at some scale qQ characteristic of quarkonia. So the matching hypothesis, supported by figs. 2 and 3, asserts a(n'MC)(qQ) = a(a)(qQ).
piing, or to determining AQCD. The bare quark mass is adjusted to obtain the spin average of the 1S states that is measured in the lab. Otherwise figs. 2 and 3 represent predictions of quenched QCD. The success of these calculations permits one to extract the basic parameters, c~a and mq. There are four steps: 1. Compute the charm- and b o t t o m o n i u m spectra with nl,MC ----0, 2 or 3 flavors of virtual quark loops. (nl,MC = 0 corresponds to the quenched approximation; hi,Me = 2 quenches just the strange quark; hi,Me = 3 would be the real world.)
3. Unless ny,MC = 3, correct for the quenched approximation. 4. Eliminate a from a~(Tr/a) and am~(Tr/a)
(11)
Potential models tell us that 200 < qc < 800 MeV for charm and 200 < qb < 1400 MeV for bottom. Step 3 yields a(nt'MC)(~r/a), so one can use the two-loop perturbative renormalization group to run from 7r/a to qQ. The perturbative running is an overestimate if qQ is taken at the lower end of these ranges. (For light hadrons, qlight ~ AQCD, so there would be no perturbative control whatso ever. ) This argument was used for the original lattice determinations of the strong coupling [4,5], and its reliability was confirmed in n/,MO ---- 2 calculations [34]. Currently the most accurate result is from ref. [7], a ? ) ( 8 . 2 GeV) = 0.196 + 0.003,
2. With perturbation theory, convert the bare lattice coupling a~'~t'Mc) to the quarkpotential (V) or MS scheme; convert the bare lattice mass (moa) ('~l,Mc) to the pole or MS scheme. The natural scale for this conversion is near (but not quite [15]) r/a.
(10)
(12)
based on ny,MC : 0 and n),,MC : 2 results, with an extrapolation in n I. The V scheme is preferred for the matching argument, not only for physical reasons, but also because of its empirical scaling behavior [15]. The scaling behavior implies that one can run with the two-loop renormalization group to high scales and convert to other schemes. For comparison to other determinations, eq. (12) corresponds to
a~-~(Mz) = 0.115 + 0.002.
(13)
276
A.X KronfeM/Nuclear Physics B (Proc. Suppl.) 49 (1996) 269-281
The quoted uncertainty is smaller than that reported from any other method. The largest contributor is the quenched correction; the second largest is the perturbative conversion 0 ~ V MS. To determine the quark mass one apples the same renormalization-gronp argument. But quark masses don't run below threshold [9]! Hence, for heavy quarks m(n"MC)(mQ) = m(na"MC)(qQ) = m(3)(qQ)= m(a)(mQ). The only corrections are perturbative, from lattice conventions to MS or physical conventions. Using the convention of the perturbative "pole mass" mc = 1.5 + 0.2 GeV (ref. [8], prehminary), (14) m b = 5.0 + 0.2 GeV (ref. [9]). (Again, for fight quarks the threshold is deep in brown muck, and all bets are off.) At the nonperturbative level confinement wipes out the pole, so the name "pole mass" should not be taken too hterally. Here the perturbative pole mass is hke a running mass, except that it has a fixed scale built into the definition. It is useful, because it is thought [7] to correspond to the mass of phenomenological models that do not probe energies less than AQCD, such as potential models. In other contexts--such as the study of Yukawa couphngs in unification scenarios--the MS convention may be more appropriate. Eq. (14b) corresponds to r n b , ~ ( m b ) = 4.0 =k 0.1 GeV. 4. T H E C K M M A T R I X Electroweak decays of flavored hadrons follow the schematic formula (experimental)= measurement known f a c t o r s ] ( factor QCD ) ( CKM factor ) "
(15)
North American, Japanese, and European taxpayers provide us with lots of money for the relevant experiments, because they want to know the CKM factors. But unless we calculate the inherently nonperturbative QCD factor, they will be sorely disappointed.
It is convenient to start with the assumption of three-generation unitarity. Then
V~,dV&+ VcdV£ + v,~v,; = 0,
(16)
an equation that prescribes a triangle in the complex plane. Dividing by VedV*b and writing VudV~*b/VcdV*b = fi + iO, one sees that unitarity predicts
~d~; V~dV£
-
1 -
f
-
if/.
(17)
The notation [35] (f, fl) is to distinguish these parameters from the standard Wolfenstein parameters (p, rl) = (~,fl)/lV,*d]. The standard CKM phase in Table 1 is 6 = tan-l(rffp) = t a n - l ( # / f ) . One would hke to determine (f, ~) through as many physical processes as possible. For example, the strength of C P violation in b-flavored hadrons is related to the angles of the triangle, hence the high interest in B factories. If all experiments agree, the test verifies the CKM explanation of C P violation; if not, the discrepancy would have to be explained by physics beyond the Standard Model. Meanwhile, lattice QCD is useful for measuring the sides of the unitarity triangle. Depending on the shape of the triangle, the precision may be good enough to predict the angles before the B factories have been built. Consider the CKM matrix elements in eq. (16). Assuming three-generation unitarity, 1 - ]Vtbl is too small to worry about, and IV~a]- IV,,, I is also very small. In principle, however, IV,,, l a n d IV¢dl can be determined from lattice QCD and measurements of the semi-leptonic decays K ---+ Trey and D ---, rrlv, respectively. The technique is the same as for IV,,b] from B ---+a'lv, discussed below. It is unlikely that lattice QCD can, or will need to, improve on IV,,dl = 0.9744+0.0010 during the period relevant to this discussion. The most poorly known elements ofeq. (16) are [Vcb[, ]Vub[, and [Vtd[. In principle, the first two can be determined from leptonic decays Bq ~ r~,, q -- c, u, but the experimental prospects are bleak. The semi-leptonic decay is more promising. Near qmax 2 = (mB -- roD*)2 the differential decay rate for B ---+D*lv is dq 2 -
factors
IAl(q2)l~lV~b[2'
(18)
277
A.S. Kronfeld/Nuclear Physics B (Proc. Suppl.) 49 (1996) 269-281
T
[l .[)"
"l
i
T
i
! 0 d P.I
~L z -- 0 5 [
:
I
t
(LEO
i
!
I 0
] '2
1 -I \'~
1 (5
Vb
0 0~
0
I
~
:l
]P~i ((;e\: 'v) Figure 4. The Isgur-Wise function ~(w) (essentially the form factor A1 of the text) from lattice QCD and CLEO. The kinematic variable O) = "Oa " Vb = 1 - (qmax 2 - q 2 ) / 2 m B m D ' ' From ref. [41].
where q2 is the invariant mass-squared of the lepton system. One must carry out a nonperturbative QCD calculation to obtain the form factor At(q2). By heavy-quark symmetry, however, A t (qmax) 2 obeys a normalization condition, up to 1/m2D. corrections [36] (estimated to be smart) and known radiative corrections. Other form factors, which are phase-space suppressed near 2 qmax, are also related by heavy-quark symmetry to A~ (q2). Hence, eq. (18) provides an essentially model-independent [37] way to determine IV~bl. The difficulty with the model-independent analysis is that the decay rate vanishes at qmax" 2 To aid experimentalists' extrapolation to that point, several groups [38-40] have used quenched lattice QCD to compute the slope of A1. A typical analysis is to fit the slope to lattice-QCD numerical data, and then fit the normalization to CLEO's experimental data, as shown in fig. 4. For example, Simone of the UKQCD Collaboration finds [41]
=
11.49 ps"
The first error is experimental; the second is from the lattice-QCD slope. Unfortunately, it is not
Figure 5. The differential rate for B ---* ~rlv as a function of pion momentum p,~. From ref. [42].
clear how to correct for the quenched approximation, and the associated uncertainty has not been estimated. Moreover, consistency checks of varying lattice spacing, volume, etc, are still in progress. Nevertheless, the overall consistency with experiment, shown in fig. 4, is encouraging. The even less well-known [Vub[ can be obtained from the semi-leptonic decays B ~ ply and B 2 nit,. Expanding in q2 near qmax = (mB - m~) ~, the differential decay rate for B ~ ulv reads dF [known] dq 2 = factors
If+(q~)121V"bl2'
(20)
where f+(q2) is the form factor that must be calculated in lattice QCD. Now, however, heavyquark symmetry does not restrict f + ( q m2a x ) , SO a calculation is needed to make any progress. These calculations are underway at Fermilab, and, presumably, many other places. Fig. 5 shows our prefiminary results [42]. In particular, note that at nonzero values of the pion momentum, the numerical lattice data are not too awful. Indeed, the available lattice momenta overlap with the the regime accessible to the CLEO experiment [43]. The third row of the CKM matrix can be probed via the box diagrams responsible for neutral meson mixing. The C P = + admixture of
A.X Kronfeld/Nuclear Physics B (Proc. Suppl.) 49 (1996) 26~28I
278
the KL is parameterized by [eKI = 2.26 x 10 -3. The Standard Model predicts
[~KI=
•e -
known ] factors
bKIV"eV"'121V~bl2× # (IVCbl2(1 - f)yttlef~.(yt)+
r~
t 16,~2m~ ~ A ( y , )
×
(23)
~ms~f~Bs~lVt*dVtbl 2, 8
(21)
y~(~A(u,) - 71)), where yq = m q2 / m 2w. This formula assumes three-generation unitarity and neglects the deviation of IV.I and IV~bl from unity. The ~/i and fi multiplying the CKM factors arise from box diagrams and their QCD corrections [44]. The nonperturbative QCD factor is -s~_t2l t K~J K2 D , K , which is the K ° - / £ ° transition matrix element of a A S = 2 operator. An encouragin[g result for the renormalizationgroup invariant BK is [45,46]
[~g = (a~g(#))-s/2SBK(NDR, #) = 0.825 -4-0.027(stat.) -t- 0.023(syst.) 4- (few %) ± 3%.
given by
(22)
The "few %" are for the quenched approximation. This estimate comes from repeating some of the numerical computations for full QCD [47], though not enough to obtain the other error bars, and from an analysis of chiral logarithms [48]. The latter study is reassuring only for degenerate quarks, so the calculations are done with both quarks at ~(m, + me). The 3% uncertainty is an estimate of O ( m , - me) contributions. But there is cause for concern. The central value quoted in eq. (22) is the result of a continuum-limit extrapolation. A sound theoretical argument [45] suggests that the extrapolation should be done hnearly in a 2. More recent numerical work [49], with more lattice spacings and smaller statistical error, is more compatible with a lattice-spacing dependence finear in a. At nonzero lattice spacing the unextrapolated values of BK agree. Unfortunately, the difference in central values that arise from the two different extrapolations is larger than all other uncertainties combined. Mixing in the B ° - / ) ° system is also sensitive to Vtd. In the Standard Model the mass sphtting is
2
2
^
The same formula holds for the B , - / ) , system, but with the d quark replaced by s (i.e. Be ~ B,, Vtd ~ Vt,:) The nonperturbative QCD factor is 8 2 2 • • ~m . B q f~ q BB~, which Is the Bq-Bq transition metrlx element of a A B = 2 operator. The calculation of the decay constant fB has received a great deal of attention over the last several years [5q], but the matrix element needed here, s 2 f ~ B B , has been mostly neglected. (There ~mB are some older, exploratory papers [51].) It is interesting to see how the lattice results influence the unitarity triangle. Fig. 6 shows constraints from eqs. (21), IV,,b/Vcbh and ~d/~,, taking for the masses r n , ~ g = 1.3 ± 0.2 GeV, mt,~g= 175 + 15 GeV,
(24)
for the hadronic matrix elements / ~ g : 0 . 8 2 5 :J= 0 . 0 5 0 ,
IfB~/fB. I ----0.90 + 0.05, I B , , / B B , I = 1.0 =k 0.2,
(25)
for "experimental"--nonperturbative QCD is needed to extract these resnltsl--CKM results
IVcbl-- 0.040 ± 0.005, 0.08 + 0.02,
IV~b/Y~bi =
(26)
and for nentral-B mixing measurements Zd = 0.72-1- 0.08,
x, = 15 -4- 5.
(27)
Other inputs are as in ref. [35]. I've made two wild guesses: ]BBe/BB, ] and z,. But note that I take the uncertainty estimate in / ~ g seriously; doubling it would not make much difference, in view of the uncertainties in mt and ]V¢bl. Alas, these and the other uncertainties are too large to make fig. 6 interesting. W h a t if lattice QCD calculation and the experiments improve? Consider for the masses m,~g=
1.3 + 0.1 GeV, mt,~-g = 175 ± 5 GeV,
(28)
A.S. Kronfeld/Nuclear Physics B (Proc. Suppl.) 49 (1996) 269 28l
279
//./ iz / /
0.5
0
0.5
i
,
,
~
-1
,
t i,
-0.5
1
,
,
,
,i . . . .
0
i
.
.
.
.
.
0.5
-1
-0.5
p
0
0.5
1
P
Figure 6. Constraints on (p,~) from I~KI (solid IV,,b/Vcbl (dashed circles with origin (0,0)), and z d / x , (dash-dotted circles with origin (1,0)), and contemporary uncertainties.
hyperbolae),
Figure 7. Constraints on (p, f/) from I~/~1 (solid hyperbolae), low IV~,b/Vcbl (dashed circles with origin (0,0)) or high IV,,b/Vcbl (dotted circles with origin (0,0)), and z d / z , (dash-dotted circles with origin (1,0)), and improved (5-10%) uncertainties.
for the hadronic matrix elements /}K = 0.825 + 0.027, I f B J f B , l = 0.90 -4- 0.02, tBB~/BB. I = 1.0 :t: 0.1,
(29)
in particular eliminating almost all the statistical error in /}K; for "experimental" CKM results
tVcsl = 0.035 d: 0.002, 0.004 "low," = 0.091 -4- 0.004 "high,"
IV,,b/V~bl---- 0.080 +
(30)
and for neutral-B mixing measurements
Xd=0.724-0.04,
x, = 18 4- 2.
(31)
Fig. 7 shows how this 5-10% level of precision improves the limits on (p, #). The wildest guess remains x,, so ignore the dashed-dotted curves momentarily. The region allowed by the hyperbolic band from $g and the circular band from [V,,b/V~b] shrinks if [V,,b] is too small. The tension between these two constraints is partly a consequence of the low value of [Vcb[ suggested by eq. (19). Increasing IV~b[ brings the hyperbolic band down more rapidly than it shrinks the circular band. If the real-world values of [Vcb[ and [V,~b/V¢b] allow a sizable region, as for the dotted circles in fig. 7, neutral-B mixing becomes crucial. The constraint becomes more restrictive as x, increases. Unfortunately, the experimental measurement becomes more difficult as x, increases.
If it proves impossible to obtain useful information on z,, one can return to eq (23), and focus on Zd alone. The lattice-QCD calculations of s _tlt2B Jt2B "° fi B will carry larger uncertainties, however, than the Ba : B, ratio. 5. C O N C L U S I O N S This talk has examined several ways in which lattice QCD can aid the determination of standard-model couplings. The quenched lattice calculations may be divided into several classes, according to the maturity of the error analysis and the presumed reliability of the quenched approximation. One class consists of the lighthadron and quarkonia spectra and the K - K mixing parameter BK. For them the straightforward uncertainties (statistics, a, L, excited states, perturbation theory) seem fairly estimated. The quenched approximation is another matter. In quarkonia, one can correct for it with potential models, yielding determinations of a s and the charm and b o t t o m masses. The quenched error in BK is also thought to be under control. Taking the error bars of ref. [45,46] at face value, B K would no longer be the limiting factor in the ]eKe constraint on the unitarity triangle. While this is, perhaps, a valid long-term view, the recent resuits ofref. [34] encourage caution. A second class
280
A.£ Kronfeld/Nuclear PhysicsB (Proc. Suppl.) 49 (1996) 26~281
consists of f9, the semi-leptonie form factors of K and D mesons (not discussed in this talk, but see ref. [50]), and the Isgur-Wise function. These quantities are essential for direct determinations of the first two rows of the CKM matrix. The quenched-approximation calculations are in good shape, but the the corrections to it cannot be simply estimated. A third class consists of the semileptonic decay B --* 7rlv and neutral-B mixing, for which only exploratory work has appeared. Nevertheless, all QCD quantities discussed here will follow a conceptually clear path to ever-moreprecise results. The next ten years or so will almost certainly witness computing and other technical improvements that will allow for wideranging calculations without the quenched approximation. By then the most efficient techniques for extracted the most relevant information will have been perfected. ACKNOWLEDGEMENTS
Fermilab is operated by Universities Research Association, Inc., under contract DE-AC0276CH03000 with the U.S. Department of Energy. REFERENCES
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Left. B345 (1995) 42. 8. A.X. E1-Khadra and B.P. Mertens, Nucl. Phys. B Proc. Suppl. 42 (1995) 406 9. C.T.H. Davies, K. Hornbostel, A. Langnau, G.P. Lepage, A. Lidsey, C.J. Morningstar, J. Shigemitsu, and J. Sloan, Phys. Rev. Left. 73 (1994) 2654. 10. S.R. Sharpe, in CP Violation and the Limits of the Standard Model, edited by J.F. Donoghue (World Scientific, Singapore, 1995); A. Soni, in Physics Beyond the Standard Model at ~he B Factory, edited by A.I. Sanda and S. Suzuki (World Scientific, Singapore, 1995). 11. G.P. Lepage, "What is Renormalization?" in From Actions to Answers, edited by T. DeGrand and D. Toussaint (World Scientific, Singapore, 1990). 12. K.G. Wilson and J. Kogut, Phys. Reports 12C (1974) 75. 13. K.G. Wilson, in Recent Developments in Gauge Theories, edited by G. 't Hooft, et al. (Plenum, New York, 1980). 14. K. Symanzik, in Recent Developments in Gauge Theories, edited by G. 't Hooft, et al. (Plenum, New York, 1980); P. Weisz, Nucl. Phys. B212 (1983) 1. 15. G.P. Lepage and P.B. Mackenzie, Phys. Rev. D48 (1993) 2250. 16. T.L. Bell and K.G. Wilson, Phys. Rev. B I I (1975) 3431; P. Hasenfratz and F. Niedermeyer, Nucl. Phys. B414 (1994) 785; U.-J. Wiese, Phys. Left. B315 (1993) 417. 17. K. Jansen, C. Liu, M. L/ischer, H. Simma, S. Sint, R. Sommer, P. Weisz, U. Wolff, DESY report DESY-95-230 (hep-lat/9512009). 18. A.S. Kronfeld, Nucl. Phys. B Proc. Suppl. 30 (1993) 445; P.B. Mackenzie, Nucl. Phys. B Proc. Suppl. nO (1993) 35; A.X. E1-Khadra, A.S. Kronfeld, and P.B. Mackenzie, to appear. 19. M. Liischer, Commun. Math. Phys. 104 (1986) 177, 105 (1986) 153. 20. F. Butler, H. Chen, J. Sexton, A. Vaccarino, and D. Weingarten, Phys. Rev. ge~t. 70 (1993) 729; gucL Phys. B430 (1994) 179.
A.S Kronfeld/Nuclear Physics B (Proc. SuppL) 49 (1996) 26~281 21. F. Butler, H. Chen, J. Sexton, A. Vaccarino, and D. Weingarten, NucI. Phys. B421 (1994) 217. 22. R. Groot, J. Hoek, and J. Smit, Nucl. Phys. B237 (1984) 111; M.F.L. Golterman and J. Smit, Phys. Lett. 1401] (1984) 392. 23. C.J. Morningstar, Phys. Rev. D48 (1993) 2265. 24. A.S. Kronfeld and B.P. Mertens, Nucl. Phys. B Proc. Suppl. 34 (1994) 495 25. J. Gasser and H. Leutwyler, Phys. Rept. 87 (1982) 77. An update is H. Leutwyler, in Perspectives in the StandaRd Model, edited by R.K. Ellis, C.T. Hill and J.D. Lykken (World Scientific, Singapore, 1992). 26. A. Ukawa, Nucl. Phys. B Proc. Suppl. 30 (1993) 3. 27. P.B. Mackenzie, Nucl. Phys. B Proc. Suppl. 34 (1994) 400. 28. C.R. Allton, M. Ciuchini, M. Crisafulli, E. Franco, V. Lubicz, and G. Martinelh, Nucl. Phys. B431 (1994) 667. 29. G.P. Lepage and B.A. Thacker, Nucl. Phys. B Proe. Suppl. 4 (1987) 199; B.A. Thacker and G.P. Lepage, Phys. Rev. D43 (1991) 196; W.E. Caswell and G.P. Lepage, Phys. Left. 167B (1986) 437; G.P Lepage, L. Magnea, C. Nakhleh, U. Magnea, and K. Hornbostel, Phys. Rev. D46 (1992) 4052. 30. E. Eichten, Nucl. Phys. B Proc. Suppl. 4 (1987) 170; E. Eichten and F. Feinberg, Phys. Rev. Left. 43 (1979) 1205; Phys. Rev. D28 (1981) 2724; E. Eichten and B. Hill, Phys. Left. B284 (1990) 511; B243 (1990) 427. 31. C.T.H. Davies, K. Hornbostel, A. Langnau, G.P. Lepage, A. Lidsey, J. Shigemitsu, and J. Sloan, Phys. Rev. DS0 (1994) 6963. 32. A.X. E1-Khadra, Nucl. Phys. B Proe. Suppl. 80 (1993) 449; A.X. E1-Khadra, G.M. Hockney, A.S. Kronfeld, P.B. Mackenzie, T. Onogi, and J.N. Simone, in progress. 33. J. Sloan, Nucl. Phys. B Proc. Suppl. 42 (1995) 171.
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34. S. Aoki, M. Fukugita, S. Hashimoto, N. Ishizuka, H. Mino, M. Okawa, T. Onogi, and A. Ukawa, Phys. Rev. Left. 74 (1995) 22. 35. A.J. Buras, M.E. Lautenbacher, and G. Ostermaier, Phys. Rev. DS0 (1994) 3433. 36. M. Luke, Phys. Lett. B252 (1990) 447. 37. M. Neubert, Phys. Lett. B264 (1991) 455. 38. C. Bernard, Y. Shen, and A. Soni Phys. Lett. B317 (1993) 164. 39. S.P. Booth, et al (UKQCD Collaboration), Phys. Rev. Lett. 72 (1994) 462. 40. J.E. Mandula and M.C. Ogilvie, e-print hep-lat/9408006. 41. J.N. Simone (UKQCD Collaboration), Nucl. Phys. B Proc. Suppl. 34 (1994) 486. 42. J.N. Simone, Fermilab report FERMILAB96/017-T (hep-ph/yymmnn~). 43. L. Gibbons, University of Rochester report UR-1427 (hep-ex/9508012). 44. T. Inami and C.S. Lira, Prog. Theo. Phys. 65 (1981) 297; (E) 65 (1981) 772. 45. S.R. Sharpe, Nucl. Phys. B Proc. Suppl. 34 (1994) 403; G.W. Kflcup, S.R. Sharpe, R. Gupta, and A. Patel, Phys. Rev. Left. 64 (1990) 25. 46. N. Ishizuka, M. Fukugita, H. Mino, M. Okawa, Y. Shizawa, and A. Ukawa, Phys. Rev. Left. 71 (1993) 24. 47. G.W. Kilcup, Phys. Rev. Left. 71 (1993) 1677. 48. S.R. Sharpe, Phys. Rev. D41 (1990) 3233; C.W. Bernard and M.F.L. Golterman, Phys. Rev. D46 (1992) 853. 49. S. Aoki, et al (JLQCD Collaboration), Tsukuba University report UTHEP-322 (hep-lat/9510012). 50. Some recent reviews with useful bibliography include ref. [10] and R. Sommer, Nucl. Phys. B Proc. Suppl. 42 (1995) 186; C.W. Bernard, Nucl. Phys. B Proc. Suppl. 34 (1994) 47. 51. C. Bernard, T. Draper, G. Hockney, and A. Soni, Phys. Rev. D38 (1988) 3540; M.B. Gavela, L. Maiani, S. Petrarca, G. Martinelli, and O. Pene, Phys. Left. 11206 (1988) 113; A. Abada et al, Nucl. Phys. B376 (1992) 172.
EI.SEVIER
Nuclear Physics B (Proc. Suppl.) 49 (1996) 269581
PROCEEDINGS SUPPLEMENTS
Lattice QCD and the standard model Andreas S. Kronfeld ~ ~Theoretieal Physics Group, Fermi National Accelerator Laboratory, Batavia, Illinois, USA Most of the poorly known parameters of the Standard Model cannot be determined without reliable calculations in nonperturbative QCD. Lattice gauge theory provides a first-principles definition of the required functional integrals, and hence offers ways of performing these calculations. This paper reviews the progress in computing hadron spectra and electroweak matrix elements needed to determine ~xs, the quark masses, and the CabibboKobayashi-Maskawa matrix.
I. I N T R O D U C T I O N Many reviews of elementary particle physics start by celebrating (or lamenting!) the success of the Standard Model. Indeed, with some eighteen or nineteen parameters the SU(3)xSU(1)xU(1) gauge theory explains an enormous array of experiments. Even a terse compendium [1] of the experiments is more than big enough to fill a phone book. A glance at Table 1 shows, however, that roughly half of the parameters are not so well determined. To test the Standard Model stringently one must learn the values of these parameters more precisely. Most of this volume contains abstract theoretical physics. Often the motivation of such work is to deduce a more appealing and more unified theory of particle physics. It might be possible to find better theories by relying on aesthetics and mathematical consistency; Einstein's general relativity is often cited as an example. Nevertheless, any natural science must have recourse to some experimental data. For a model based on superstrings, or whatever else, one could take the view that Table 1 represents the data. Then it is alarming that many of the most central data are known only within a factor of three or four. Under these circumstances, it will be difficult to know if beauty found in mathematics is the same as beauty found in nature. Except for the mass of the Higgs boson (or any other undiscovered remnant of electroweak symmetry breaking), the poorly known parameters all involve quarks. Other than top [2], which de-
cays too quickly for confinement to play a role, the masses of the quarks are a bit better than wild guesses. The information on the CabibboKobayashi-Maskawa (CKM) quark-mixing matrix is spotty, especially when one relaxes the assumption of three-generation unitarity, as shown in Table 2. They are poorly determined simply because experiments measure properties not of quarks, but of the hadrons inside which they are confined. Of course, everyone knows what to do: calculate with QCD, the part of the Standard Model that describes the strong interactions. But the strong coupfing a s is known only at the 5% level; not bad, but nothing fike the fine structure or Fermi constants. Moreover, the binding of quarks into hadrons is nonperturbative-even knowing o~s, hadronic calculations cannot be done on the back of an envelope. The most systematic technique for understanding nonperturbative QCD is lattice gauge theory. The lattice provides quantum field theory with a consistent and mathematically well-defined ultraviolet regulator. At fixed lattice spacing, the quantities of interest are straightforward (combinations of) functional integrals. These integrals can be approximated by a variety of techniques borrowed from statistical mechanics. Especially promising is a numerical technique, the Monte Carlo method with importance sampling, which has become so pre-eminent that the young and uninitiated probably haven't heard of any other. Results from lattice-QCD Monte Carlo calculations have begun to influence Table 1. The world average for the SU(3) gauge coupling a s includes
0920-5632/96/$15.00 -~ 1996 Elsevicr Scicnce B.V. All rights reserved. PII: S0920-5632(96)00344-1
270
A.S. KronfeM/Nuclear Physics B (Proc. Suppl.) 49 (1996) 26~281
Table 1 Parameters of the standard model and lattice calculations that will help determine them. Numerical values taken from the 1994 Review of Particle Properties [1], except m t (ref. [2]), sin 6, and 0QCD. The strong coupling a s refers to the MS scheme at M z . Adapted from ref. [3]. what
value or range
how
gauge couplings O%m
1/137.036 1.166 GeV -2 0.116 + 0.005
105GF as electroweak m a s s e s mz 91.19 GeV
AmlP-lS
what
value or range
IV ,, l
0.974
IVy,, [
0.218-0.224 0.002-0.005 0.180-0.228 0.800-0.975 0.032-0.048 0.0-0.13
IVy,b[ [Vca[
]Vc,[ [V~s[
> 58 GeV
IV,al Iv.I
0.0-0.56
0.51100 MeV 105.66 MeV 1777 MeV
IV, hi
0.0-0.9995
mH lepton m a s s e s
me mu m~.
Table 2 Ranges for CKM matrix elements [Vq~] assuming unitarity but not three generations. Numerical values taken from the 1994 Review of Particle Properties. In three generations IVy,d[ = s12, Iv~bl -- s=3, and [Vub[ = 813, to excellent approximation. how K --* Trey B --, flu D --, f l y D ---, K l u B -~ D * l v f~BB; BK
f .Bs.
quark m a s s e s
rn~ m d
m, mc
2-8 MeV 5-15 MeV 100-300 MeV 1.0-1.6 GeV 4.1-4.5 GeV 180 ± 14 GeV
2 m~, m ~ m~
0.218-0.224 0.032-0.048 0.002-0.005
K ---* ~rev B --* D* lu B ---* ~rlv BK, BB, BB.
mb mt CKM matrix
st2 s2a sta sin 6
#0
msl¢
mi
Q C D v a c u u m angle
0QCD
< 10 -9
dn
results from lattice calculations of the quarkonium spectrum [4-7]. The same calculations are also providing some of the best information on the charm [8] and b o t t o m [9] masses. This is an auspicious beginning. Over the next several years the lattice Q C D calculations will mature. They will help to determine the other unknowns--light quark masses and the C K M matrix. The third column of Tables I and 2 listsrelevant quantities or processes, and the rest of this talk explains how
the program fits together. Sect. 2 gives the non-expert some Perspective on the conceptual and numerical strengths and weaknesses of lattice QCD. Sect. 3 reviews 1) the status of the light hadron spectrum and the prospects for extracting mu, rod, and m , ; and 2) results for the quarkonium spectrum, which yield a s , rob, and me. Sect. 4 outlines lattice QCD calculations of eleetroweak, hadronic matrix elements that are needed to pin down the unitarity triangle of the CKM matrix. There are, of course, m a n y other interesting applications to electroweak phenomenology; for more comprehensive reviews the reader can consult some of the papers listed in the bibliography [3,10]. 2. R U D I M E N T S
OF LATTICE
GAUGE
THEORY In quantum field theory physical measurements are related to vacuum expectation values ((9). Feynman's functional integral representation is
((9)= L--oolim=--0nmZ-lfHaA"(~)× *," H d¢~ (z)d¢~(z) (9 e - s ( a " ' ¢ ' • ) ,
(1)
271
A.S. KronfeM/Nuclear Physics B (Proc. Suppl.) 49 (1996) 26~281
where S is the action--in our case the action of QCD, so it depends on the gluons A~ and the quarks ¢ and anti-quarks ¢. The observable (9 depends on the physics under investigation. The integration over all components and positions of the basic fields, with weight e i s would reproduce the familiar SchrSdinger or Heisenberg formulations of quantum mechanics. The more convergent weight e - s provides some benefits and imposes some restrictions--see below. The limits are there for mathematical rigor; without them the integrals are not well defined. These "cutoffs" also have a physical significance: we do not claim to understand physics either at distances smaller than a (the ultraviolet cutoff), or at distances larger than L (the infrared cutoff). The limit a ~ 0 requires the renormalization group; it must be carried out holding L and physical, infrared scales fixed. In particular, the integration variables At,(z), ¢ ( z ) , and ¢ ( z ) really represent all degrees of freedom in a block of size a 4. The limit "a ---* 0" can be obtained not only hterally, but also by improving the action of the blocked fields. These observations apply to any cutoff scheme for quantum field theory. A nice introduction to the renormalization-group aspects is a summerschool lecture by Lepage [11]. In lattice gauge theory a is nothing but the spacing between lattice sites. If there are N on a side, L = N a . For given N one can compute the integrals numerically. With the 10z-101°-dimensional integrals that arise, the only viable technique is a statistical one: Monte Carlo with importance samphng. In the last few years Monte Carlo calculations have improved in m a n y ways. An especially keen development is the use of improved actions. By universality any action
(2) n
with arbitrary interactions S,~ built with the right symmetry out of the appropriate fields, should have the same continuum bruit. But the cutoff artifacts can be reduced systematically by choosing the "irrelevant" couplings as functions of the "relevant" one. In the case of QCD the relevant coupfings are the gauge couphng and (dimension-
less) quark masses. A so-called renormalized trajectory c°
(Z)
--
has no cutoff effects. Making an error 6Cn in the couphng of the Monte Carlo's action leads to artifacts .-~ 6c,~(S,~), so the objective is to estimate the eouphngs cn of the most important interactions S,~. Of course, this strategy is old, going back to Wilson [12,13] and to Symanzik [14]. New are more effective ways [15-17] of computing the couphngs cn(g2,rnqa), so that cutoff artifacts are genuinely reduced. In particular, for massive quarks, such as charm and bottom, it is wise [18] to treat exactly the mass dependence of the renormahzed trajectory prescribed by eq. (3). With good estimates of the c,~, the cutoff effects can be reduced below other relevant uncertainties. To compute masses one takes an observable O = ff(t)ff*(0), where if(t) is an operator at time t with the flavor and angular-momentum quantum numbers of the state of interest. One can eonstruet such operators using s y m m e t r y alone. (The radial quantum number would require a solution of the theory, but that's what we're after.) The two-point function (~(t)~*(0))
=~
(4)
e - m " ' I(01~l n) 12 ,
n
where the sum is over the radial q u a n t u m number. The exponentials are a happy consequence of the weight e - s in eq. (1). It is advantageous because at long times t only the lowestlying state survives. In a numerical calculation masses are obtained by fitting two-point functions, once single-exponential behavior is verified. Since • is largely arbitrary, some artistry enters: if single-exponential behavior sets in sooner, the statistical quality of the mass estimate is better. Most of Table 1 requires a matrix element of part of the electroweak Hamiltonian 7-/. For the transition from hadron "B" to hadron "Tr" one employs the observable O = ¢b~(t,~)7~(th)fft(o). At long times th and t , - t h the three-point function (ff.(t,)~(th)~(0))
~
e -m'(t'-t~)-rnBta
(01¢. I~r)0rlT~lB)(Bl~ 10),
x
(5)
A.S. Kronfeld/Nuclear Physics B (Proc. Suppl.) 49 (1996) 26~281
272
plus excited-state contributions. If, as in decays of hadrons to leptons, the hadronic final-state is the vacuum, a two-point function will do:
(~(t)O~(O)) = ~ The
e-~"t(O['}/[B,~)(B,~IO~]O>.
(6)
desired
matrix elements (~r[7~[B) and be obtained from eq. (5) and (6), because the masses and matrix elements of eh are obtained from eq. (4). To obtain good results from eqs. (4)-(6), it is crucial to devise nearly optimal operators in the two-point analysis. Consumers of numerical results from lattice QCD should be wary of results, still too prevalent in the literature, that are contaminated by unwanted states. Fortunately the goal of improving the hadronic part of Table 1 can be met with two- and threepoint functions only. Four-point functions are much more demanding technically. In the numerical work that mostly concerns us here, the integrals are computed at a sequence of fixed a's and L's. One adopts a standard mass, say rap, and defines
(oI~IB> can
masses. From the renormalization group, the adjustment of the gauge coupling is equivalent to setting the lattice spacing in physical units, eq. (7). Once the parameters are determined by 1 + n! experimental inputs, QCD should predict all other strong-interaction phenomena. There is no need to introduce condensates, as in I T E P sum rules, or non-renormalizable couplings, as in chiral perturbation theory or heavy-quark effective theory. If theory and experiment disagree, it is a signal of new physics. There are some disadvantages. A practical, though not conceptual, problem is that largescale computational work is more labor-intensive than traditional theoretical physics. Careful work is needed to estimate the uncertainties reliably. Recent improvements in computer power and algorithms have helped practitioners understand their uncertainties better and better. As the consumers of their results become commensurately sophisticated, this trend will continue. After all, in the context of Table 1, meaningful error bars are just as important as the central value.
(amp) 1QOD a -
oxp,
(7)
mp
to obtain the lattice spacing in physical units, and other quantities are predicted via (amB)lq cD mB
-
(s)
a
For continuum-limit, infinite-volume results this is the same as extrapolating dimensionless ratios,
mB mp
lim lira L--oo a - - O
amp(a, L)
.
(9)
There is theoretical guidance for both limits. According to general properties of massive quant u m field theories in finite boxes [19], the infinitevolume limit is rapid for m,~L >> 1, exponential for masses. In non-Abelian gauge theories the renormalization-group a ---* 0 limit is controlled by asymptotic freedom. The main strength of lattice QCD is that it is QCD. It has 1 + n! adjustable parameters, corresponding to the gauge coupling and the quark
2.1. T h e q u e n c h e d a p p r o x i m a t i o n The biggest disadvantage of most of the numerical results mentioned in this talk is something called the "quenched" approximation. A meson consists of a valence quark and anti-quark exchanging any number ofgluons. The gluons can turn into virtual quark loops and back again. The latter process costs a factor of 100-1000 in computer time, so many Monte Carlo programs just omit the virtual quark loops. Borrowing a jargon from condensed-matter physics, this is most often called the quenched approximation. If quenched QCD makes any sense, it is as a kind of model or effective theory. The parameters of quenched QCD can be tuned to reproduce physics at one scale. But the/3 function of quenched and genuine QCD differ, as one sees in perturbation theory, so one cannot expect agreement at all scales. As with any model, only in special cases can one argue that these effects are correctable or negligible; these cases will be highlighted in the rest of the talk.
A.S. KronfeM/Nuclear Physics B (Proc. Suppl.) 49 (1996) 26~281 2.5 GFI 1 experiment
t
t.
ft
t
1.5
t" i
I
K* 0
I
L
I
I
i
N-+E-NA Z* E* f~ 10fitlOfKfK/~t~t
Figure 1. The spectrum and decay constants of the hght hadrons. Error bars are from lattice calculations in the quenched approximation [20,21], and • denotes experiment.
3. Q C D C O U P L I N G S : MASSES
as AND
QUARK
3.1. L i g h t h a d r o n s a n d l i g h t - q u a r k m a s s e s Over the past few years a group at IBM has carried out a systematic calculation of the hghthadron spectrum using the dedicated supercomputer G F l l [20]. They have numerical d a t a for 5 different combinations of (a, L). At L m, 2.3 fm there are three lattice spacings varying by a factor of ~ 2.5. At the coarsest lattice spacing (a -1 ~ 1.4 GeV) there are three volumes, up to almost 2.5 fm. A variety of quark masses are used, and the physical strange quark is reached by interpolation, but the hght (up and down) quarks through extrapolation. The mass dependence is assumed linear, as expected from weakly broken chiral symmetry, and the d a t a substantiate the assumption. Units in MeV have been fixed with mp and the quark masses with m~ and m g . The final results, after extrapolation to the continuum limit and infinite volume, are shown in fig. 1 for two vector mesons and six baryons. (The quark-mass interpolation could reach only the combination mE + m s - mN.) Despite the quenched approximation the agreement with experiment is spectacular.
273
Fig. 1 also includes results from the same investigation for decay constants. The agreement of f,~/mp and fIc/mp is not as good as for the masses. Because of the quenched approximation, this is not entirely unexpected. Recall the argument concerning distance scales and effective theories in sect. 2.1. The binding mechanism responsible for the masses encompasses distances out to the typical hadronic radius. The decay constant, on the other hand, is proportional to the wavefunetion at the origin and thus is more sensitive to shorter distance scales. One sees better agreement when forming the ratio fg/f,~, which--recall eq. (7) and subsequent discussion-is like retuning to the shorter distance. One would like to use the hadron masses to extract the quark masses. Because of confinement, the quark mass more like a renormallzed couphng than the classical concept of mass. Calculations like the one described above yield immediately the bare mass of the lattice theory. More useful to others would be the MS scheme of dimensional regularization. A one-loop perturbative calculation can be used to convert from one scheme to another [22-24,8]. For the light quarks it is convenient to discuss the combinations ~ = ~(md + rn~), Am~,, = m ] - m~, 2 and ms Ratios of the light-quark masses are currently best estimated using chiral perturbation theory [25]. To set the overall scale requires a dynamical calculation in QCD. In lattice QCD, ~h and m, can be extracted from the variation in the square of the pseudoscalar mass between m r2 and m~c. The most difficult quark-mass combination is A m ~ , which causes the isospin-violating contribution to the sphttings in hadron multiplets. Since chiral perturbation theory provides a formula for Am~,/m2, with only second-order corrections, it is likely that the best determination of A m ~ will come from combining the formula with a lattice QCD result for m~. Using the compilation of quenched and unquenched results of Ukawa [26], Mackenzie [27] has estimated thug(1 GeV) ~ 2.3 MeV and m ,~-g(1 GeV) ,~ 65 MeV. The symbol -~ stresses the lack of error bar. This is outside the ranges of 3.5-11.5 MeV and 100-300 MeV indicated in Table 1. A more recent analysis of the strange
274
A.X Kronfeld/Nuclear Physics B (Proc. Suppl.) 49 (1996) 269 281
quark finds m , ~ ( 2 GeV) = 127 ± 18 MeV [28], in the lower part of the range in Table 1. None of these results should be taken seriously until a more complete error analysis exists, but it is intriguing that the conventional estimates might be too high. Q u a r k o n i a : a s , me, a n d m b Quarkonia are bound states of a heavy quark and heavy anti-quark. Three families of states exist, charmonium (7/¢, J / ¢ , etc), bottomonium (~/b, T, etc), and the as yet unobserved Be (b~ and bc bound states). Compared to light hadrons, these systems are simple. The quarks are nonrelativistie, and potential models give an excellent empirical description. But a fundamental treatment of these systems requires nonperturbative QCD, i.e. lattice QCD. Potential models can be exploited, however, to estimate lattice artifacts, and in the quenched approximation they can be used to make corrections. Many states have been observed in the lab, providing cross-checks of the methodology of uncertainty estimation. Once the checks are satisfactory, one can use the spectra to determine a s , m~, and ms. One can also have some confidence in further applications, such as the phenomenology of D and B mesons discussed in sect. 4. For charm, and especially for bottom, the quark mass is close to the ultraviolet cutoff, 1/a or ~r/a, of present-day numerical calculations. Originally lattice gauge theory was formulated assuming rnqa << 1, so quarks mqa ~-- 1 require some reassessment. There are four ways to react. The patient, stolid way is to wait ten years, until computers are powerful enough to reach a cutoff of 20 G e V - - n o t very inspiring. The naive way is to extrapolate from smaller masses, assuming the mqa << 1 interpretation of the lattice theory is adequate; history shows that naive extrapolations can lead to naive and, thus, unacceptable error estimates. The insightful way is to formulate an effective theory for heavy quarks with a lattice cutoff [29,30]; this is the computationally most efficient approach, and when the effectiveness of the heavy-quark expansion is a priori clear, it is the method of choice. The compulsive way to examine a wide class of lattice theories without 3.2.
> CD 0.5
-
0.0
-
D3
I
77c
J/~
hc
Xc0
Xcl
Xcz
Figure 2. A comparison of the charmonium spectrum as calculated in lattice QCD, using two different methods, ref. [31]: o, ref. [32]: D. From
ref. [6].
assuming either mqa << 1 or mq >> (AQcD, a - l ) ; by imposing physical normalization conditions on masses and matrix elements, it is possible to interpret the correlation functions at any value of mqa [18]. The underlying reason is that the lattice theory is completely compatible with the heavy-quark hmit, so the mass-dependent interpretation connects smoothly onto both the insightful method for mqa >> 1 and the standard method for mqa << 1. Fig. 2 shows the charmonium spectrum, on a scale appropriate to the spin-averaged spectrum. Light quark loops are quenched in these calculations [31,32]. The agreement with experimental measurements is impressive, but fig. 2 barely displays the attainable precision. Fig. 3 shows the fine and hyperfine structure of the P states, now for bottomonium [31]. (The 1P1 state hb has not been observed in the lab; the hc has been seen.) The authors of ref. [31] also have results with the virtual quark loops from two light quarks, i.e. up and down are no longer quenched, but strange still is. The agreement is comparable [33]. To obtain these results only two parameters have been adjusted. The standard mass in eq. (7) is AmlP-lS, the spin-averaged splitting of the 1P and 1S states, which is insensitive to the quark mass. By the renormahzation group, this is equivalent to ehminating the bare gauge cou-
A.S. Kronfeld/Nuclear Physics B (Proc. Suppl.) 49 (1996) 269 281 MeV
using
20--+--Xb2 a-0
* hb
--~--Xbl
--20 --40
275
--t-- Xbo
Figure 3. Lattice QCD results for the spindependent splittings of the lowest-lying P states in bottomonium. The dashed lines are the experimental values, where available. Energies are measured relative to the spin average of the states Xbj" From ref. [31].
aAmlP-lS 460 MeV '
where the numerator is the 1P-1S splitting in lattice units. Steps 1 and 4 are explained above. Step 2 requires one-loop perturbation theory, suitably optimized [15]. Step 3 is crucial, because without it the results have no business in Table 1. Consider first a s , and recall the idea of treating the quenched approximation as an effective theory. One sees that the couplings are implicitly matched at some scale qQ characteristic of quarkonia. So the matching hypothesis, supported by figs. 2 and 3, asserts a(n'MC)(qQ) = a(a)(qQ).
piing, or to determining AQCD. The bare quark mass is adjusted to obtain the spin average of the 1S states that is measured in the lab. Otherwise figs. 2 and 3 represent predictions of quenched QCD. The success of these calculations permits one to extract the basic parameters, c~a and mq. There are four steps: 1. Compute the charm- and b o t t o m o n i u m spectra with nl,MC ----0, 2 or 3 flavors of virtual quark loops. (nl,MC = 0 corresponds to the quenched approximation; hi,Me = 2 quenches just the strange quark; hi,Me = 3 would be the real world.)
3. Unless ny,MC = 3, correct for the quenched approximation. 4. Eliminate a from a~(Tr/a) and am~(Tr/a)
(11)
Potential models tell us that 200 < qc < 800 MeV for charm and 200 < qb < 1400 MeV for bottom. Step 3 yields a(nt'MC)(~r/a), so one can use the two-loop perturbative renormalization group to run from 7r/a to qQ. The perturbative running is an overestimate if qQ is taken at the lower end of these ranges. (For light hadrons, qlight ~ AQCD, so there would be no perturbative control whatso ever. ) This argument was used for the original lattice determinations of the strong coupling [4,5], and its reliability was confirmed in n/,MO ---- 2 calculations [34]. Currently the most accurate result is from ref. [7], a ? ) ( 8 . 2 GeV) = 0.196 + 0.003,
2. With perturbation theory, convert the bare lattice coupling a~'~t'Mc) to the quarkpotential (V) or MS scheme; convert the bare lattice mass (moa) ('~l,Mc) to the pole or MS scheme. The natural scale for this conversion is near (but not quite [15]) r/a.
(10)
(12)
based on ny,MC : 0 and n),,MC : 2 results, with an extrapolation in n I. The V scheme is preferred for the matching argument, not only for physical reasons, but also because of its empirical scaling behavior [15]. The scaling behavior implies that one can run with the two-loop renormalization group to high scales and convert to other schemes. For comparison to other determinations, eq. (12) corresponds to
a~-~(Mz) = 0.115 + 0.002.
(13)
276
A.X KronfeM/Nuclear Physics B (Proc. Suppl.) 49 (1996) 269-281
The quoted uncertainty is smaller than that reported from any other method. The largest contributor is the quenched correction; the second largest is the perturbative conversion 0 ~ V MS. To determine the quark mass one apples the same renormalization-gronp argument. But quark masses don't run below threshold [9]! Hence, for heavy quarks m(n"MC)(mQ) = m(na"MC)(qQ) = m(3)(qQ)= m(a)(mQ). The only corrections are perturbative, from lattice conventions to MS or physical conventions. Using the convention of the perturbative "pole mass" mc = 1.5 + 0.2 GeV (ref. [8], prehminary), (14) m b = 5.0 + 0.2 GeV (ref. [9]). (Again, for fight quarks the threshold is deep in brown muck, and all bets are off.) At the nonperturbative level confinement wipes out the pole, so the name "pole mass" should not be taken too hterally. Here the perturbative pole mass is hke a running mass, except that it has a fixed scale built into the definition. It is useful, because it is thought [7] to correspond to the mass of phenomenological models that do not probe energies less than AQCD, such as potential models. In other contexts--such as the study of Yukawa couphngs in unification scenarios--the MS convention may be more appropriate. Eq. (14b) corresponds to r n b , ~ ( m b ) = 4.0 =k 0.1 GeV. 4. T H E C K M M A T R I X Electroweak decays of flavored hadrons follow the schematic formula (experimental)= measurement known f a c t o r s ] ( factor QCD ) ( CKM factor ) "
(15)
North American, Japanese, and European taxpayers provide us with lots of money for the relevant experiments, because they want to know the CKM factors. But unless we calculate the inherently nonperturbative QCD factor, they will be sorely disappointed.
It is convenient to start with the assumption of three-generation unitarity. Then
V~,dV&+ VcdV£ + v,~v,; = 0,
(16)
an equation that prescribes a triangle in the complex plane. Dividing by VedV*b and writing VudV~*b/VcdV*b = fi + iO, one sees that unitarity predicts
~d~; V~dV£
-
1 -
f
-
if/.
(17)
The notation [35] (f, fl) is to distinguish these parameters from the standard Wolfenstein parameters (p, rl) = (~,fl)/lV,*d]. The standard CKM phase in Table 1 is 6 = tan-l(rffp) = t a n - l ( # / f ) . One would hke to determine (f, ~) through as many physical processes as possible. For example, the strength of C P violation in b-flavored hadrons is related to the angles of the triangle, hence the high interest in B factories. If all experiments agree, the test verifies the CKM explanation of C P violation; if not, the discrepancy would have to be explained by physics beyond the Standard Model. Meanwhile, lattice QCD is useful for measuring the sides of the unitarity triangle. Depending on the shape of the triangle, the precision may be good enough to predict the angles before the B factories have been built. Consider the CKM matrix elements in eq. (16). Assuming three-generation unitarity, 1 - ]Vtbl is too small to worry about, and IV~a]- IV,,, I is also very small. In principle, however, IV,,, l a n d IV¢dl can be determined from lattice QCD and measurements of the semi-leptonic decays K ---+ Trey and D ---, rrlv, respectively. The technique is the same as for IV,,b] from B ---+a'lv, discussed below. It is unlikely that lattice QCD can, or will need to, improve on IV,,dl = 0.9744+0.0010 during the period relevant to this discussion. The most poorly known elements ofeq. (16) are [Vcb[, ]Vub[, and [Vtd[. In principle, the first two can be determined from leptonic decays Bq ~ r~,, q -- c, u, but the experimental prospects are bleak. The semi-leptonic decay is more promising. Near qmax 2 = (mB -- roD*)2 the differential decay rate for B ---+D*lv is dq 2 -
factors
IAl(q2)l~lV~b[2'
(18)
277
A.S. Kronfeld/Nuclear Physics B (Proc. Suppl.) 49 (1996) 269-281
T
[l .[)"
"l
i
T
i
! 0 d P.I
~L z -- 0 5 [
:
I
t
(LEO
i
!
I 0
] '2
1 -I \'~
1 (5
Vb
0 0~
0
I
~
:l
]P~i ((;e\: 'v) Figure 4. The Isgur-Wise function ~(w) (essentially the form factor A1 of the text) from lattice QCD and CLEO. The kinematic variable O) = "Oa " Vb = 1 - (qmax 2 - q 2 ) / 2 m B m D ' ' From ref. [41].
where q2 is the invariant mass-squared of the lepton system. One must carry out a nonperturbative QCD calculation to obtain the form factor At(q2). By heavy-quark symmetry, however, A t (qmax) 2 obeys a normalization condition, up to 1/m2D. corrections [36] (estimated to be smart) and known radiative corrections. Other form factors, which are phase-space suppressed near 2 qmax, are also related by heavy-quark symmetry to A~ (q2). Hence, eq. (18) provides an essentially model-independent [37] way to determine IV~bl. The difficulty with the model-independent analysis is that the decay rate vanishes at qmax" 2 To aid experimentalists' extrapolation to that point, several groups [38-40] have used quenched lattice QCD to compute the slope of A1. A typical analysis is to fit the slope to lattice-QCD numerical data, and then fit the normalization to CLEO's experimental data, as shown in fig. 4. For example, Simone of the UKQCD Collaboration finds [41]
=
11.49 ps"
The first error is experimental; the second is from the lattice-QCD slope. Unfortunately, it is not
Figure 5. The differential rate for B ---* ~rlv as a function of pion momentum p,~. From ref. [42].
clear how to correct for the quenched approximation, and the associated uncertainty has not been estimated. Moreover, consistency checks of varying lattice spacing, volume, etc, are still in progress. Nevertheless, the overall consistency with experiment, shown in fig. 4, is encouraging. The even less well-known [Vub[ can be obtained from the semi-leptonic decays B ~ ply and B 2 nit,. Expanding in q2 near qmax = (mB - m~) ~, the differential decay rate for B ~ ulv reads dF [known] dq 2 = factors
If+(q~)121V"bl2'
(20)
where f+(q2) is the form factor that must be calculated in lattice QCD. Now, however, heavyquark symmetry does not restrict f + ( q m2a x ) , SO a calculation is needed to make any progress. These calculations are underway at Fermilab, and, presumably, many other places. Fig. 5 shows our prefiminary results [42]. In particular, note that at nonzero values of the pion momentum, the numerical lattice data are not too awful. Indeed, the available lattice momenta overlap with the the regime accessible to the CLEO experiment [43]. The third row of the CKM matrix can be probed via the box diagrams responsible for neutral meson mixing. The C P = + admixture of
A.X Kronfeld/Nuclear Physics B (Proc. Suppl.) 49 (1996) 26~28I
278
the KL is parameterized by [eKI = 2.26 x 10 -3. The Standard Model predicts
[~KI=
•e -
known ] factors
bKIV"eV"'121V~bl2× # (IVCbl2(1 - f)yttlef~.(yt)+
r~
t 16,~2m~ ~ A ( y , )
×
(23)
~ms~f~Bs~lVt*dVtbl 2, 8
(21)
y~(~A(u,) - 71)), where yq = m q2 / m 2w. This formula assumes three-generation unitarity and neglects the deviation of IV.I and IV~bl from unity. The ~/i and fi multiplying the CKM factors arise from box diagrams and their QCD corrections [44]. The nonperturbative QCD factor is -s~_t2l t K~J K2 D , K , which is the K ° - / £ ° transition matrix element of a A S = 2 operator. An encouragin[g result for the renormalizationgroup invariant BK is [45,46]
[~g = (a~g(#))-s/2SBK(NDR, #) = 0.825 -4-0.027(stat.) -t- 0.023(syst.) 4- (few %) ± 3%.
given by
(22)
The "few %" are for the quenched approximation. This estimate comes from repeating some of the numerical computations for full QCD [47], though not enough to obtain the other error bars, and from an analysis of chiral logarithms [48]. The latter study is reassuring only for degenerate quarks, so the calculations are done with both quarks at ~(m, + me). The 3% uncertainty is an estimate of O ( m , - me) contributions. But there is cause for concern. The central value quoted in eq. (22) is the result of a continuum-limit extrapolation. A sound theoretical argument [45] suggests that the extrapolation should be done hnearly in a 2. More recent numerical work [49], with more lattice spacings and smaller statistical error, is more compatible with a lattice-spacing dependence finear in a. At nonzero lattice spacing the unextrapolated values of BK agree. Unfortunately, the difference in central values that arise from the two different extrapolations is larger than all other uncertainties combined. Mixing in the B ° - / ) ° system is also sensitive to Vtd. In the Standard Model the mass sphtting is
2
2
^
The same formula holds for the B , - / ) , system, but with the d quark replaced by s (i.e. Be ~ B,, Vtd ~ Vt,:) The nonperturbative QCD factor is 8 2 2 • • ~m . B q f~ q BB~, which Is the Bq-Bq transition metrlx element of a A B = 2 operator. The calculation of the decay constant fB has received a great deal of attention over the last several years [5q], but the matrix element needed here, s 2 f ~ B B , has been mostly neglected. (There ~mB are some older, exploratory papers [51].) It is interesting to see how the lattice results influence the unitarity triangle. Fig. 6 shows constraints from eqs. (21), IV,,b/Vcbh and ~d/~,, taking for the masses r n , ~ g = 1.3 ± 0.2 GeV, mt,~g= 175 + 15 GeV,
(24)
for the hadronic matrix elements / ~ g : 0 . 8 2 5 :J= 0 . 0 5 0 ,
IfB~/fB. I ----0.90 + 0.05, I B , , / B B , I = 1.0 =k 0.2,
(25)
for "experimental"--nonperturbative QCD is needed to extract these resnltsl--CKM results
IVcbl-- 0.040 ± 0.005, 0.08 + 0.02,
IV~b/Y~bi =
(26)
and for nentral-B mixing measurements Zd = 0.72-1- 0.08,
x, = 15 -4- 5.
(27)
Other inputs are as in ref. [35]. I've made two wild guesses: ]BBe/BB, ] and z,. But note that I take the uncertainty estimate in / ~ g seriously; doubling it would not make much difference, in view of the uncertainties in mt and ]V¢bl. Alas, these and the other uncertainties are too large to make fig. 6 interesting. W h a t if lattice QCD calculation and the experiments improve? Consider for the masses m,~g=
1.3 + 0.1 GeV, mt,~-g = 175 ± 5 GeV,
(28)
A.S. Kronfeld/Nuclear Physics B (Proc. Suppl.) 49 (1996) 269 28l
279
//./ iz / /
0.5
0
0.5
i
,
,
~
-1
,
t i,
-0.5
1
,
,
,
,i . . . .
0
i
.
.
.
.
.
0.5
-1
-0.5
p
0
0.5
1
P
Figure 6. Constraints on (p,~) from I~KI (solid IV,,b/Vcbl (dashed circles with origin (0,0)), and z d / x , (dash-dotted circles with origin (1,0)), and contemporary uncertainties.
hyperbolae),
Figure 7. Constraints on (p, f/) from I~/~1 (solid hyperbolae), low IV~,b/Vcbl (dashed circles with origin (0,0)) or high IV,,b/Vcbl (dotted circles with origin (0,0)), and z d / z , (dash-dotted circles with origin (1,0)), and improved (5-10%) uncertainties.
for the hadronic matrix elements /}K = 0.825 + 0.027, I f B J f B , l = 0.90 -4- 0.02, tBB~/BB. I = 1.0 :t: 0.1,
(29)
in particular eliminating almost all the statistical error in /}K; for "experimental" CKM results
tVcsl = 0.035 d: 0.002, 0.004 "low," = 0.091 -4- 0.004 "high,"
IV,,b/V~bl---- 0.080 +
(30)
and for neutral-B mixing measurements
Xd=0.724-0.04,
x, = 18 4- 2.
(31)
Fig. 7 shows how this 5-10% level of precision improves the limits on (p, #). The wildest guess remains x,, so ignore the dashed-dotted curves momentarily. The region allowed by the hyperbolic band from $g and the circular band from [V,,b/V~b] shrinks if [V,,b] is too small. The tension between these two constraints is partly a consequence of the low value of [Vcb[ suggested by eq. (19). Increasing IV~b[ brings the hyperbolic band down more rapidly than it shrinks the circular band. If the real-world values of [Vcb[ and [V,~b/V¢b] allow a sizable region, as for the dotted circles in fig. 7, neutral-B mixing becomes crucial. The constraint becomes more restrictive as x, increases. Unfortunately, the experimental measurement becomes more difficult as x, increases.
If it proves impossible to obtain useful information on z,, one can return to eq (23), and focus on Zd alone. The lattice-QCD calculations of s _tlt2B Jt2B "° fi B will carry larger uncertainties, however, than the Ba : B, ratio. 5. C O N C L U S I O N S This talk has examined several ways in which lattice QCD can aid the determination of standard-model couplings. The quenched lattice calculations may be divided into several classes, according to the maturity of the error analysis and the presumed reliability of the quenched approximation. One class consists of the lighthadron and quarkonia spectra and the K - K mixing parameter BK. For them the straightforward uncertainties (statistics, a, L, excited states, perturbation theory) seem fairly estimated. The quenched approximation is another matter. In quarkonia, one can correct for it with potential models, yielding determinations of a s and the charm and b o t t o m masses. The quenched error in BK is also thought to be under control. Taking the error bars of ref. [45,46] at face value, B K would no longer be the limiting factor in the ]eKe constraint on the unitarity triangle. While this is, perhaps, a valid long-term view, the recent resuits ofref. [34] encourage caution. A second class
280
A.£ Kronfeld/Nuclear PhysicsB (Proc. Suppl.) 49 (1996) 26~281
consists of f9, the semi-leptonie form factors of K and D mesons (not discussed in this talk, but see ref. [50]), and the Isgur-Wise function. These quantities are essential for direct determinations of the first two rows of the CKM matrix. The quenched-approximation calculations are in good shape, but the the corrections to it cannot be simply estimated. A third class consists of the semileptonic decay B --* 7rlv and neutral-B mixing, for which only exploratory work has appeared. Nevertheless, all QCD quantities discussed here will follow a conceptually clear path to ever-moreprecise results. The next ten years or so will almost certainly witness computing and other technical improvements that will allow for wideranging calculations without the quenched approximation. By then the most efficient techniques for extracted the most relevant information will have been perfected. ACKNOWLEDGEMENTS
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